Whisk together the lemon juice, olive oil, mustard, rosemary, garlic and a big pinch of salt and pepper in a measuring cup. Add chicken to a gallon-size zip top bag or casserole dish and pour the marinade over the chicken. Turn to coat and let the chicken marinate at room temperature for 30 minutes (or 1 hour in the refrigerator).
Heat a grill to medium-high. Brush the grill grates with olive oil using a clean tea towel or paper towel.
Remove the chicken from the marinade and discard the marinade. Place the chicken on the grill and cook until golden and slightly charred, about 3 minutes per side. Place the lemon halves cut-side down on the grill grate and grill until charred, about 4 minutes.
Add the arugula to a platter and drizzle very lightly with olive oil. Remove the chicken from the grill and shingle across the bed of arugula. Squeeze some lemon juice over the chicken and serve.
Recipe courtesy of Jamie Deen